A Kawamata-Viehweg Vanishing Theorem on compact K¨ ahler manifolds

A Kawamata-Viehweg Vanishing Theorem on compact K¨ ahler manifolds

Jean-Pierre Demailly^⋆, Thomas Peternell^⋆⋆

⋆ Universit´e de Grenoble I, BP 74 ^⋆⋆ Universit¨at Bayreuth Institut Fourier, UMR 5582 du CNRS Mathematisches Institut 38402 Saint-Martin d’H`eres, France D-95440 Bayreuth, Deutschland

Abstract. We prove a Kawamata-Viehweg vanishing theorem on a normal compact K¨ahler space X: if L is a nef line bundle with L² 6= 0, then H^q(X, K_X + L) = 0 for q ≥ dimX −1. As an application we complete a part of the abundance theorem for minimal K¨ahler threefolds: if X is a minimal K¨ahler threefold, then the Kodaira dimension κ(X) is nonnegative.

§0. Introduction

In this paper we establish the following Kawamata-Viehweg type vanishing theorem on a compact K¨ahler manifold or, more generally, a normal compact K¨ahler space.

0.1 Theorem. Let X be a normal compact K¨ahler space of dimension n and L a nef line bundle on X. Assume that L² 6= 0. Then

H^q(X, K_X+L) = 0 for q≥n−1.

In general, one expects a vanishing

H^q(X, KX+L) = 0

for q ≥ n+ 1−ν(L), where ν(L) is the numerical Kodaira dimension of the nef line bundle L, i.e. ν(L) is the largest integer ν such that L^ν 6= 0.

Of course, when X is projective, Theorem 0.1 is contained in the usual Kawamata- Viehweg vanishing theorem, but the methods of proof in the algebraic case clearly fail in the general K¨ahler setting. Instead we proceed in the following way. Clearly we may assume thatX is smooth and by Serre duality, only the cohomology group Hⁿ⁻¹ is of interest. Take a singular metric h on L with positive curvature current T with local weight function h. By [Si74, De93a] there exists a decomposition

T =X

λ_jD_j+G, (D)

where λj ≥ 1 are irreducible divisors, and G is a pseudo-effective current such that G|D_i is pseudo-effective for alli. Consider the multiplier ideal sheafI(h). We associate to h another, “upper regularized” multiplier ideal sheaf I₊(h) by setting

I₊(h) := lim

ε→0+

I(h^1+ε) = lim

ε→0+

I (1 +ε)ϕ .

It is unknown whether I(h) and I₊(h) actually differ; in all known examples they are equal. Then in Section 2 the following vanishing theorem is proved.

0.2 Theorem. Let (L, h) be a holomorphic line bundle over a compact K¨ahler n-fold X. Assume that L is nef and has numerical Kodaira dimension ν(L) =ν >0, i.e. c₁(L)^ν 6= 0 and ν is maximal. Then the morphism

H^q(X,O(K_X +L)⊗I₊(h))−→H^q(X, K_X+L) induced by the inclusion I₊(h)⊂O_X vanishes for q > n−ν.

The strategy of the proof of Theorem 0.2 is based on a direct application of the Bochner technique with special hermitian metrics constructed by means of the Calabi- Yau theorem.

Now, coming back to the principles of the proof of Theorem 0.1, we introduce the divisor

D =X

[λj]Dj.

Then Theorem 0.2 yields the vanishing of the map in cohomology Hⁿ⁻¹(X,−D+L+K_X)−→Hⁿ⁻¹(X, L+K_X).

Thus we are reduced to show that Hⁿ⁻¹(D, L+K_X|D) = 0, or dually that H⁰(D,−L+D|D) = 0.

This is now done by a detailed analysis of a potential non-zero section in −L+D|D;

making use of the decomposition (D) and of a Hodge index type inequality.

The vanishing theorem 0.1 is most powerful when X is a threefold, and in the second part of the paper we apply 0.1 - or rather a technical generalization - to prove the following abundance theorem.

0.3 Theorem. Let X be a Q-Gorenstein K¨ahler threefold with only terminal singu- larities, such that K_X is nef (a minimal K¨ahler threefold for short). Then κ(X)≥0.

This theorem was established in the projective case by Miyaoka and in [Pe01] for K¨ahler threefolds, with the important exception that X is a simple threefold which is not Kummer. Recall that X is said to be simple if there is no proper compact subvariety through a very general point ofX, and that X is said to be Kummer ifX is bimeromorphic to a quotient of a torus. So our contribution here consists in showing that such a simple threefoldX withK_X nef has actuallyκ(X) = 0. Needless to say that among all K¨ahler threefolds the simple non-Kummer ones (which conjecturally do not exist) are most difficult to deal with, since they do not carry much global information besides the fact that π1 is finite and that they have a holomorphic 2-form.

The first main ingredient in our approach is the inequality K_X·c₂(X)≥0

§1. Preliminaries 3

for a minimal simply connected K¨ahler threefoldX with algebraic dimensiona(X) = 0.

Philosophically this inequality comes from Enoki’s theorem that the tangent sheaf of X is KX-semi-stable when K_X² 6= 0 resp. (K_X, ω)-semi-stable when K_X² = 0; here ω is any K¨ahler form on X. Now if this semi-stability with respect to a degenerate polarization would yield a Miyaoka-Yau inequality, then KX·c2(X)≥0 would follow.

However this type of Miyoka-Yau inequalities with respect to degenerate polarizations is completey unknown. In the projective case, the inequality follows from Miyaoka’s generic nefness theorem and is based on char. p-methods. Instead we approximate KX (in cohomology) by K¨ahler forms ωj. If TX is still ωj-semi-stable for sufficiently large j, then we can apply the usual Miyaoka-Yau inequality and pass to the limit to obtain KX ·c2(X) ≥0. Otherwise we examine the maximal destabilizing subsheaf which essentially (because of a(X) = 0) is independent of the polarization.

The second main ingredient is the boundedness h²(X, mK_X) ≤ 1. If K_X² 6= 0, this is of course contained in Theorem 0.1. If K_X² = 0, we prove this boundedness under the additional assumption that a(X) = 0 and that π₁(X) is finite (otherwise by a result of Campana X is already Kummer). The main point is that if h²(X, mKX)≥2, then we obtain “many” non-split extensions

0−→K_X −→E−→ mK_X −→0

and we analyze whether E is semi-stable or not. The assumption on π₁ is used to conclude that if E is projectively flat, then E is trivial after a finite ´etale cover.

From these two ingredients Theorem 0.3 immediately follows by applying Riemann- Roch on a desingularization of X.

The only remaining problem concerning abundance on K¨ahler threefolds is to prove that a simple K¨ahler threefold with K_X nef and κ(X) = 0 must be Kummer.

§1. Preliminaries

We start with a few preliminary definitions.

1.1 Definition. A normal complex space X is said to be K¨ahler if there exists a K¨ahler form ω on the regular part of X such that the following holds. Every singular point x ∈ X admits an open neighborhood U and a closed embedding U ⊂ V into an open set U ⊂C^N such that there is a K¨ahler form η on V with η|U =ω.

1.2 Remark. Let X be a compact K¨ahler space and let f : ˆX −→X be a desingu- larization by a sequence of blow-ups. Then ˆX is a K¨ahler manifold. More generally consider a holomorphic map f : ˆX −→ X of a normal compact complex space to a normal compact K¨ahler space. If f is a projective morphism or, more generally, a K¨ahler morphism, then ˆX is K¨ahler. For references to this and more informations on K¨ahler spaces, we refer to [Va84].

A K¨ahler form ω defines naturally a class [ω] ∈ H²(X,R), see [Gr62] where K¨ahler metrics on singular spaces were first introduced. Therefore we also have a K¨ahler cone on a normal variety.

1.3 Notation. Let X be a normal compact complex space.

(1) Let A and B be reflexive sheaves of rank 1. Then we define A⊗Bˆ := (A⊗B)^∗∗. Moreover we let A^[m] :=A^⊗mˆ .

(2) A reflexive sheaf A is said to be a Q-line bundle if there exists a positive integer m such that A^[m] is locally free.

(3)X is Q-Gorenstein if the canonical reflexive sheafωX, also denoted KX, is a Q-line bundle. X is Q-factorial, if every reflexive sheaf of rank 1 is aQ-line bundle.

1.4 Definition. Let X be a normal compact K¨ahler threefold.

(1)X is simple if there is no proper compact subvariety through the very general point of X.

(2)X is Kummer, if X is bimeromorphic to a quotientT /G where T is a torus andG a finite group acting on T.

It is conjectured that all simple threefolds are Kummer.

1.5 Notation.

(1) The algebraic dimensiona(X) of an irreducible reduced compact complex space is the transcendence degree of the field of meromorphic functions over C. If a(X) = 0, i.e. all meromorphic functions on X are constant, then it is well known thatX carries only finitely many irreducible hypersurfaces.

(2) A line bundle L on a compact K¨ahler manifold is nef, if c₁(L) lies in the closure of the K¨ahler cone. For alternative descriptions see e.g. [DPS94,00]. If X is a normal compact K¨ahler, thenL is nef if there exists a desingularizationπ: ˆX −→X such that π^∗(L) is nef. By [Pa98], this definition does not depend on the choice of π.

§2. Hodge index type inequalities

We give here some generalizations of Hodge index inequalities for nef classes over compact K¨ahler manifolds. In this direction the main result is the Hovanskii-Teissier concavity inequality, which can be stated in the following way (see e.g. [De93b], Prop.

5.2 and Remark 5.3).

2.1 Proposition. Let α1, . . . , αk and γ1, . . . γn−k be nef cohomology classes on a compact K¨ahler n-dimensional manifold X. Then

α₁· · ·α_k·γ₁· · ·γ_n−k >(α^k₁ ·γ₁· · ·γ_n−k)^1/k· · ·(α^k_k·γ₁· · ·γ_n−k)^1/k.

We want to derive from these a non vanishing property for intersection products of the form αⁱ·β^j. Let us fix a K¨ahler metric ω on X. By Proposition 2.1 applied with k =i+j and theα_ℓ’s beingicopies ofα followed by j copies ofβ andγ_ℓ =ω, we have

αⁱ·β^j·ω^n−i−j >(α^k·ω^n−k)^i/k· · ·(β^k·ω^n−k)^j/k.

§2. Hodge index type inequalities 5

As all products α^k and analogues can be represented by closed positive currents, we have α^k 6= 0⇒α^k·ω^n−k>0, hence with k =i+j we find

(2.2) α^i+j 6= 0 and β^i+j 6= 0 =⇒ αⁱ·β^j 6= 0.

This is of course optimal in terms of the exponents if α = β, but as we shall see in a moment, this is possibly not optimal in a dissymetric situation. Actually, we have the following additional inequalities which can be viewed as “differentiated” Hovanskii- Teissier inequalities.

2.3 Theorem. Let α and β be nef cohomology classes of type (1,1) on a compact K¨ahler n-dimensional manifold X. Assume that α^p 6= 0 and β^q 6= 0 for some integers p, q >0. Then we have αⁱ·β^j 6= 0as soon as there exists an integerk >i+j such that

i(k−p)₊+j(k−q)₊ < k, where x₊ means the positive part of a number x.

Proof. Assume thatαⁱ·β^j = 0. We apply the Hovanskii-Teissier inequality respectively with αℓ =α+εω (i terms), or αℓ =β+εω (j terms) or αℓ =ω (k−i−j terms), and γ_ℓ =ω. This gives

(∗) (α+εω)ⁱ·(β+εω)^j·ω^n−i−j > (α+εω)^k·ω^n−ki/k

(β+ε)^k·ω^n−kj/k

(ωⁿ)^{1−i/k−j/k}. By expanding the intersection form and using the assumption αⁱ·β^j = 0, we infer

(α+εω)ⁱ·(β+εω)^j ·ω^n−i−j 6O(ε)

asε tends to zero. On the other hand (α+εω)^k·ω^n−k is bounded away from 0 ifk6 p since then α^k 6= 0, and (α+εω)^k·ω^n−k > Cε^k−p for some constant C > 0 if k > p.

Hence we infer from (∗) that

Cε^{(i/k)(k−p)+}+(j/k)(k−q)+

=O(ε),

and this is not possible if i(k−p)₊+j(k−q)₊ < k. The theorem is proved.

The special casep = 2,q = 1,i =j = 1,k = 2 provides the following result which will be needed later on several occasions.

2.4 Corollary. Assume that α, β are nef with α² 6= 0 and β 6= 0. Then α·β 6= 0.

Finally, we state an extension of Proposition 2.1 in the case when one of the coho- mology classes involved is not necessarily nef.

2.5 Proposition.Let α be a real (1,1)-cohomology class, and let β, γ₁, . . . γ_n−2 be nef cohomology classes. Then

(α·β·γ₁· · ·γ_n−2)² >(α²·γ₁· · ·γ_n−2)(β²·γ₁· · ·γ_n−2).

Proof.. By proposition 2.1, the result is true when α is nef. If we replace β by β+εω and let ε > 0 tend to zero, we see that it is enough to consider the case when β is a

K¨ahler class. Then α+λβ is also K¨ahler for λ ≫ 1 large enough, and the inequality holds true with α +λβ in place of α. However, after making the replacement, the contributions of terms involvingλ in the right and left hand side of the inequality are both equal to

2λ(α·β·γ₁· · ·γ_n−2)(β²·γ₁· · ·γ_n−2) +λ²(β²·γ₁· · ·γ_n−2)². Hence these terms cancel and the claim follows.

§3. Partial vanishing for multiplier ideal sheaf cohomology

Let (L, h) be a holomorphic line bundle over a compact K¨ahler n-fold X. Locally in a trivializationL_|U ≃U×C, the metric is given bykξk_x =|ξ|e^−ϕ(x) and we assume that the curvature Θ_h(L) := _πⁱ∂∂ϕ is a closed positive current (so that, in particular, L is pseudo-effective). We introduce as usual the multiplier ideal sheaf I(h) := I(ϕ) where

I(ϕ)_x :=

f ∈O_X,x; ∃V ∋x, Z

V

|f(z)|²e^−2ϕ(z)<+∞

andV is an arbitrarily small neighborhood ofx. We also consider theupper regularized multiplier ideal sheaf

I₊(h) := lim

ε→0+

I(h^1+ε) = lim

ε→0+

I (1 +ε)ϕ . It should be noticed that I (1 +ε)ϕ

increases as ε decreases, hence the limit is lo- cally stationary by the Noether property of coherent sheaves, and one has of course I₊(h)⊂I(h). It is unknown whether these sheaves may actually differ (in all known examples they are equal). In any case, they coincide at least in codimension 1 (i.e., outside an analytic subset of codimension>2).

3.2 Proposition. Let

Θh(L) =

+∞

X

j=1

λjDj +G

be the Siu decomposition of the (1,1)-current Θh(L) as a countable sum of effective divisors and of a (1,1)-current G such that the Lelong sublevel sets E_c(G), c > 0, all have codimension 2. Then we have the inclusion of sheaves

I₊(h)⊂I(h)⊂O −X

[λ_j]D_j

, [λ_j] :=integer part of λ_j,

and equality holds on XrZ where Z is an analytic subset ofX whose components all have codimension at least 2.

Proof.. The decomposition exists by [Siu74] (see also [De93a]). Now, if g_j is a local generator of the ideal sheafO(−D_j), the plurisubharmonic weightϕofhcan be written as

ϕ=X

λ_jlog|g_j|+ψ

§3. Partial vanishing for multiplier ideal sheaf cohomology 7

where ψ is plurisubharmonic and the Ec(ψ) have codimension 2 at least. Since ψ is locally bounded from above, it is obvious that

I(ϕ)⊂I(λ_jlog|g_j|)⊂O −X

[λ_j]D_j .

Now, letY be the union of all sets E_c(ψ) (with, say, c= 1/k), all pairwise intersections D_j∩D_kand all singular setsD_jsing. This setY is at most a countable union of analytic sets of codimension>2. Pick an arbitrary pointx∈XrY. Thenxmeets the support of S

D_j in at most one point which is then a smooth point of some D_k, and the Lelong number of ψ_k = ψ+P

j6=kλ_jlog|g_j| at x is zero. Then ϕ = λ_klog|g_k|+ψ_k and the inclusion

I(h)x ⊃O −X

j

[λj]Dj

x =O(−[λ_k]Dk)x

holds true by H¨older’s inequality. In fact, for every germ f in O(−[λ_k]Dk)x we have Z

V∋x

|f|²exp −(1 +ε)λ_klog|g_k|

<+∞

for ε > 0 so small that [(1 +ε)λk] = [λk], while e^−ψk is in L^p(Vp) for some Vp ∋x, for every p >1. Similarly, we have

I₊(h)_x ⊃O −[(1 +ε)λ_k]D_k

x =O −[λ_k]D_k

x

for ε >0 small enough. The analytic set Z where our sheaves differ

i.e. the union of supports ofI(h)/I₊(h) and O −P

j[λj]Dj /I(h)

must be contained inY, hence Z is of codimension>2.

The main goal of this section is to prove the following partial vanishing theorem.

3.3 Theorem. Let (L, h) be a holomorphic line bundle over a compact K¨ahler n-fold X, equipped with a singular metric h such that Θ_h(L) > 0. Assume that L is nef and has numerical dimension ν(L) = ν >0, i.e. c₁(L)^ν 6= 0 and ν is maximal.

Then the morphism

H^q(X,O(K_X +L)⊗I₊(h))−→H^q(X, K_X+L) induced by the inclusion I₊(h)⊂O_X vanishes for q > n−ν.

Of course, it is expected that the Kawamata-Viehweg vanishing theorem also holds for K¨ahler manifolds, in which case the whole group H^q(X, K_X + L) vanishes and Theorem 3.3 would then be an obvious consequence. However, we will see in Section 4 that, conversely, Theorem 3.3 can be used to derive the Kawamata-Viehweg vanishing theorem in the first non trivial case ν = 2. Using the same method for higher values of ν would probably be very hard, if not impossible.

Proof.. Our strategy is based on a direct application of the Bochner technique with special hermitian metrics constructed by means of the Calabi-Yau theorem.

Let us fix a smooth hermitian metric h∞ on L, which may have a curvature form Θ_h∞(L) of arbitrary sign, and let ε >0. Then c₁(L) +εω is a K¨ahler class, hence by

the Calabi-Yau theorem for complex Monge-Amp`ere equations there exists a hermitian metric h_ε =h_∞e^−2ϕε such that

(3.4) Θhε(L) +εωn

=Cεωⁿ. Here C_ε >0 is the constant such that

Cε = R

X(c₁(L) +εω)ⁿ R

Xωⁿ >Cε^n−ν.

Leth =h∞e^−2ψ be a metric with Θh(L)>0 as given in the statement of the theorem, and let ψ_ε↓ ψ be a regularization of ψ possessing only analytic singularities (i.e. only logarithmic poles), such that

h˜_ε:=h_∞e^−2ψε

satisfies Θ˜hε(L) > −εω in the sense of currents. Such a metric exists by the general regularization results proved in [De92]. We consider the metric

hˆ_ε= (h_ε)^δ(˜h_ε)^1−δ=h_∞exp −2(δϕ_ε+ (1−δ)ψ_ε)

where δ >0 is a sufficiently small number which will be fixed later. By construction, Θ_ˆh

ε(L) + 2εω =δ Θ_hε(L) +εω

+ (1−δ) Θ_˜h

ε(L) +εω +εω

>δ Θ_hε(L) +εω +εω.

Denote by λ₁ 6 . . . 6 λ_n and ˆλ₁ 6 . . . 6 λˆ_n, respectively, the eigenvalues of the curvature forms Θ_hε(L) +εω and Θ_ˆh

ε(L) + 2εω at every point z ∈X, with respect to the base K¨ahler metric ω(z). By the minimax principle we find ˆλ_j > δλ_j +ε. On the other hand, the Monge-Amp`ere equation (3.4) tells us that

(3.5) λ₁. . . λ_n =C_ε >Cε^n−ν

everywhere on X. We apply the basic Bochner-Kodaira inequality to sections of type (n, q) with values in the hermitian line bundle (L,ˆh_ε). As the curvature eigenvalues of Θˆhε(L) are equal to ˆλ_j−2ε by definition, we find

(3.6) k∂uk²_ˆh

ε +k∂^⋆uk²_ˆh

ε >

Z

X

(ˆλ₁+· · ·+ ˆλ_q−2qε)|u|²_hˆ

εdV_ω

for every smooth (n, q)-form u with values in L. Actually this is formally true only if the metric ˆh_ε is smooth on X. The metrich_ε is indeed smooth, but ˜h_ε may have poles along an analytic set Zε ⊂ X. In that case, we apply instead the inequality to forms u which are compactly supported in X rZ_ε, and replace the K¨ahler metric ω by a sequence of complete K¨ahler metricsωk ↓ω onXrZε, and pass to the limit ask tends to +∞ (see e.g. [De82] for details about such techniques). In the limit we recover the same estimates as if we were in the smooth case, and we therefore allow ourselves to ignore this minor technical problem from now on.

§3. Partial vanishing for multiplier ideal sheaf cohomology 9

Now, let us take a cohomology class {β} ∈ H^q(X, KX ⊗L ⊗I₊(h)). By using Cech cohomology and the De Rham-Weil isomorphism between ˇˇ Cech and Dolbeault cohomology (via a partition of unity and the usual homotopy formulas), we obtain a representative β of the cohomology class which is a smooth (n, q)-form with values in L, such that the coefficients of β lie in the sheaf I₊(h)⊗O_X C^∞. We want to show that β is a boundary with respect to the cohomology group H^q(X, K_X ⊗L). This group is a finite dimensional Hausdorff vector space whose topology is induced by the L² Hilbert space topology on the space of forms (all Sobolev norms induce in fact the same topology on the level of cohomology groups). Therefore, it is enough to show that we can approach β by ∂-exact forms in L² norm.

As in H¨ormander [H¨o65], we write every form u in the domain of the L²-extension of ∂^∗ as u=u₁+u₂ with

u₁ ∈Ker∂ and u₂ ∈(Ker∂)^⊥ = Im∂^∗ ⊂ Ker∂^∗. Therefore, since β ∈Ker∂,

hhβ, uii

2 =

hhβ, u₁ii

2 6 Z

X

1

ˆλ₁+· · ·+ ˆλ_q|β|²_hˆ

εdV_ω Z

X

(ˆλ₁+· · ·+ ˆλ_q)|u₁|²_hˆ

εdV_ω. As ∂u₁ = 0, an application of (3.6) to u₁ (together with an approximation of u₁ by compactly supported smooth sections on the corresponding complete K¨ahler manifold XrZε) shows that the second integral in the right hand side is bounded above by

k∂^∗u1k²_ˆh

ε + 2qεku₁k²_ˆh

ε 6k∂^∗uk²_ˆh

ε+ 2qεkuk²_ˆh

ε, so we finally get

hhβ, uii

2 6 Z

X

1

λˆ₁+· · ·+ ˆλ_q|β|²_ˆh

εdV_ω k∂^∗uk²_ˆh

ε + 2qεkuk²_ˆh

ε

.

By the Hahn-Banach theorem (or rather a Hilbert duality argument in this situation), we can find elements v_ε, w_ε such that

hhβ, uii=hhv_ε, ∂^∗uii+hhw_ε, uii ∀u, i.e. β =∂v_ε+w_ε, with

kv_εk²_ˆh

ε + 1

2qεkw_εk²_ˆh

ε 6 Z

X

1

ˆλ₁+· · ·+ ˆλ_q|β|²_hˆ

εdV_ω.

As a consequence, the L² distance of β to the space of ∂-exact forms is bounded by kw_εk_ˆh

ε where kw_εk²_ˆh

ε = Z

X

|w_ε|²_h∞e^{−2(δϕε+(1−δ)ψε)}dV_ω 62qε Z

X

1

ˆλ₁+· · ·+ ˆλ_q|β|²_ˆh

εdV_ω. We normalize the choice of the potentials ϕ_ε, ψ andψ_ε so that

sup

X

ϕ_ε = 0, sup

X

ψ=−1, −16sup

X

ψ_ε <0 ;

in this way ϕε, ψε 6 0 everywhere on X (all inequalities can be achieved simply by adding suitable constants). From this we infer

Z

X

|w_ε|²_h∞dVω 62 Z

X

qε

ˆλ₁+· · ·+ ˆλ_q|β|²_ˆh

εdVω,

and what remains to be shown is that the right hand side converges to 0 for a suitable choice of δ >0. By construction ˆλ_j >δλ_j +ε and (3.5) implies

λ^q_qλ_q+1. . . λ_n >λ₁. . . λ_n>Cε^n−ν, hence

1 λ1+· · ·+λq

6 1

λq

6C^−1/qε^{−(n−ν)/q}(λ_q+1. . . λ_n)^1/q. We infer

γ_ε:= qε

λˆ₁+· · ·+ ˆλ_q 6min 1, qε

δλ_q

6min 1, Cδ⁻¹ε^{1−(n−ν)/q}(λ_q+1. . . λ_n)^1/q . We notice that

Z

X

λ_q+1. . . λ_ndV_ω 6 Z

X

(Θ_hε(L) +εω)^n−q∧ω^q= (c₁(L) +ε{ω})^n−q{ω}^q 6C^′′, hence the functions (λ_q+1. . . λ_n)^1/q are uniformly bounded in L¹ norm as ε tends to zero. Since 1−(n−ν)/q > 0 by hypothesis, we conclude that γε converges almost everywhere to 0 as ε tends to zero. On the other hand

|β|²_ˆh

ε =|β|²_h∞e^{−2(δϕε+(1−δ)ψε)} 6|β|²_h∞e^−2δϕεe^−2ψ.

Our assumption that the coefficients of β lie in I₊(h) implies that there exists p^′ > 1 such that R

X|β|²_h∞e^−2p′ψdV_ω < ∞. Let p ∈ ]1,+∞[ be the conjugate exponent such that ¹_p + _p^1′ = 1. By H¨older’s inequality, we have

Z

X

γε|β|²_ˆh

εdVω 6 Z

X

|β|²_h∞e^−2pδϕεdVω

1/pZ

X

γ_ε^p′|β|²_h∞e^−2p′ψdVω

1/p^′

. Asγ_ε61, the Lebesgue dominated convergence theorem shows that

Z

X

γ_ε^p′|β|²_h∞e^−2p′ψdV_ω

converges to 0 asε tends to 0. However, the family of quasi plurisubharmonic functions (ϕ_ε) is a bounded family with respect to the L¹ topology on the space of (quasi)- plurisubharmonic functions – we use here the fact that the currents

Θ˜hε(L) = Θ_h∞(L) + i

π∂∂ϕ_ε>0

all sit in the same cohomology class; the boundedness of their normalized potentials then results from the continuity properties of the Green operator. By standard results of complex potential theory, we conclude that there exists a small constant η > 0 such that R

Xe^−2ηϕεdV_ω is uniformly bounded; in fact this follows from Prop. 7.1 of [Sk72], p. 389, and from the arguments of its proof. By choosing δ 6η/p, the integral R

X|β|²_h∞e^−2pδϕεdV_ε remains bounded and we are done.

§4. Kawamata-Viehweg vanishing theorem for line bundles of numerical dimension 2 11

§4. Kawamata-Viehweg vanishing theorem for line bundles of numerical dimension 2

In this section we prove the Kawamata-Viehweg vanishing theorem for the coho- mology group of degree n−1 of nef line bundles L with L² 6= 0 on compact K¨ahler spaces of dimension n. Furthermore we will prove an extended version whereL can be a reflexive sheaf. This will be needed for proving the abundance theorem for K¨ahler threefolds.

4.1 Theorem. Let X be a normal compact K¨ahler space of dimension n and L a nef line bundle on X. Assume that L² 6= 0. Then

H^q(X, K_X+L) = 0 for q≥n−1.

Proof. In a first step we reduce the proof to the case of a smooth space X (this is comfortable but not really necessary; all arguments would also work in the singular setting as well). In fact, letπ : ˆX −→X be a K¨ahler desingularization. Then, assuming our claim in the smooth case, we have

H^q( ˆX, K_Xˆ +π^∗(L)) = 0.

By the projection formula and the Grauert-Riemenschneider vanishing theorem R^jπ∗(KXˆ) = 0,

it follows

H^q(X, π_∗(K_Xˆ)⊗L) = 0.

Sinceπ_∗(KXˆ) ⊂K_X with cokernel supported in codimension at least 2, namely on the singular locus of X, the vanishing claim follows.

So from now on, we assume X smooth. In the case q =n, we haveHⁿ(X, KX +L) = H⁰(X,−L)^∗ by Serre duality, and for L nef, −L has no section unless L is trivial.

Therefore the only interesting case is q = n−1. We introduce a singular metric h on L with positive curvature current T. By [Siu74] and [De92, De93a] we obtain a decomposition

T =X

λjDj+G,

whereλ_j ≥1 are irreducible divisors, andGis a positive current such thatGhas Lelong numbers in codimension > 2 only – so that in particular G|D_i is pseudo-effective for all i. Consider the multiplier ideal sheaf I(h). By Proposition 3.2 we have

I₊(h)⊂I(h)⊂O_X(−X

[λ_j]D_j) with equality in codimension 1. We put

D =X

[λ_j]D_j.

We consider the canonical map in cohomology

Hⁿ⁻¹(X,−D+L+KX)−→Hⁿ⁻¹(X, L+KX)

which is vanishing by (3.3). In order to prove our claim it is therefore sufficient to prove

Hⁿ⁻¹(D, L+K_X|D) = 0.

By Serre duality and the adjunction formula, this comes down to show H⁰(D,−L+D|D) = 0.

Supposing the contrary, we fix a non-zero section σ ∈H⁰(D,−L+D).

We choose p1, . . . , pk maximal so that

σ ∈H⁰(D,−L+D−X

pjDj), i.e. we choose ˜D = P

p_jD_j ⊂ D maximal such that σ|D˜ = 0.

In this notation, we view ˜D as the subscheme ofX defined by the structure sheaf O_X/O_X(−D)˜

.

Then 0 ≤p_i ≤[λ_i] for all i ∈ I, not all p_i = [λ_i], and we shall always consider σ as a section of −L+D−P

p_jD_j. Denote

c_i = {λ_i}+pi

λ_i . Then we have 0≤ci ≤1. We introduce c= minci and

I₀ ={i ∈I|c_i =c}.

Clearly c <1. Notice that by construction σ|D_i6= 0 unless c_i = 1. Let E =− X

({λ_i}+p_i)D_i

−G.

Since L=P

λ_iD_i+G, we have

−L+D−X

piDi =− X

({λ_i}+pi)Di

−G=E,

so E is effective (possibly zero) on every D_i with c_i < 1. Since L is nef, also the R-divisorcL=P

λ_icD_i+cGis nef. Adding this to the divisor E in the last equation, we deduce that

− X

({λ_i}+p_i−cλ_i)D_i

−(1−c)G

is pseudo-effective on every D_i with c_i < 1. Since {λ_i}+p_i −cλ_i = 0 for i ∈ I₀, it follows that

− X

i6∈I0

({λ_i}+p_i−cλ_i)D_i

−(1−c)G

§4. Kawamata-Viehweg vanishing theorem for line bundles of numerical dimension 2 13

is pseudo-effective on every Di with ci < 1, in particular for every i∈ I0. Now Dj|D_i is effective (possibly 0) for all j 6=i, andG|D_i is always pseudo-effective, hence, having in mind c <1 and {λ_i}+pi −cλi >0 for i6∈I0, we conclude that

D_j|D_i ≡0 (1)

for all (j, i) with j 6∈I₀ and i∈I₀ and that

G|D_i ≡0 (2)

for all i ∈I₀. Introducing

D^′ = X

i∈I0

λiDi

and

D^′′ = X

i6∈I0

λ_iD_i, we have

L=D^′+D^′′+G

andD^′′·D_i =G·D_i = 0 for all i∈I₀ by (1) and (2). HenceL·D_i =D^′·D_i fori ∈I₀, so that D^′|D_i is nef, hence D^′ is nef by [Pa98]. In total

L·D^′ =D^′·D^′ and

D^′ ·D^′′ =D^′·G= 0.

AsL² 6= 0 andD^′ 6= 0, Corollary 2.4 implies L·D^′ 6= 0. First recall that L|D_i = X

j∈I0

λ_jD_j|D_i

is nef for i∈I₀. On the other hand

−X

j∈I0

{λ_j}+p_j)D_j =−cX

j∈I0

λ_jD_j

is of course pseudo-effective on every D_i for i∈ I₀ (E is effective on those D_i). Com- bining these two facts, we deduce that either c = 0 or that L·Di = 0 for all i ∈ I0, hence L·D^′ = 0, contradiction. So we have c= 0. This means p_j = 0 and λ_j ∈N for all j ∈I0.

4.2 Claim. The divisor D^′′+G is nef, and in fact must be equal to zero.

Proof. of the claim. We consider the closed positive (1,1)-current Θ = [D^′′] +G. By the results of [Pa98], the proof of nefness of {Θ} just amounts to showing that the restriction{Θ}_|Z of the (1,1)-cohomology class{Θ}to any component Z in the Lelong sublevel setS

c>0E_c(Θ) is nef. HoweverZ is either a component ofD^′′ or a component

ofS

c>0Ec(G). In the first case,Zis contained in the support ofD^′′, and asD^′·D^′′ = 0, Z must be disjoint from D^′. Hence

{Θ}_|Z ={D^′′+G}_|Z ={L−D^′}_|Z ={L}_|Z is nef. If Z is a component of S

c>0E_c(G), then Z has codimension at least 2. Then we know by [De93a] that the intersection product [D^′]∧G is well defined as a closed positive current. Since the cohomology class of this current is zero, we must have [D^′]∧G= 0. However, we infer from [De93a] that

ν([D^′]∧G, z)>ν([D^′], z)ν(G, z)>0

at every point z ∈ D^′ ∩Z, hence Z must also be disjoint from D^′ in that case. We conclude as before that {Θ}_|Z = {L}_|Z is nef. Now we have D^′ ·(D^′′ +G) = 0, with D^′, D^′′+G nef and D^′2 =L·D^′ 6= 0. Hence {D^′′ +G}= 0 by Corollary 2.4, and we conclude that D^′′ = 0, G= 0 (both [D^′′] and [G] being positive currents).

From this we infer L≡D^′ and I(h) =O_X(−D^′).

Case 1 We assume thatL =D^′. Now the sequence

0−→I(h)⊗K_X +L−→K_X +L−→K_X +L|D^′ =K_D^′ −→0 gives in cohomology

0−→Hⁿ⁻¹(X, K_X+L)−→Hⁿ⁻¹(D^′, K_D^′)≃H⁰(D^′,O_D′)−→Hⁿ(K_X) =C−→0.

Thus we need to show

h⁰(D^′,O_D′) = 1.

In order to verify this, we first observe that D^′ is connected. In fact otherwise write D^′ =A+B with A and B effective and A·B = 0. But A and B are necessarily nef, hence the Hodge Index Theorem gives a contradiction to L² = (D^′)² 6= 0. So D^′ is connected and if h⁰(D^′,O_D′) ≥ 2, then O_D′ contains a nilpotent section t 6= 0. Let P

j∈Iµ_jD_j denote its vanishing divisor (notice that D^′ is Cohen-Macaulay!). Then 1≤µj ≤λj for all j. Let

J ={j ∈J|λ_j µj

maximal}

and letc= ^λ_µ^j

j be the maximal value. Notice that−P

j∈I µ_jD_j|D_i is effective (possibly 0) for all i. First we rule out the case that c = ^λ_µ^j

j for all j ∈ I. In fact, then L|D_i = cP

µ_jD_j|D_i is nef and its dual is effective, hence L|D_i ≡ 0 for all i, whence L² = 0, contradiction. Thus we find some j such that

c > λ_j µ_j.

By connectedness of D = D^′ we can choose i₀ ∈ J in such a way that there exists j1 ∈I \J with Di0 ∩Dj1 6=∅. Now

X

j∈I

(λj−cµj)Dj|D_i0

§4. Kawamata-Viehweg vanishing theorem for line bundles of numerical dimension 2 15

is pseudo-effective as a sum of a nef and an effective line bundle (this has nothing to do with the choice of i₀). Since the sum, taken over I, is the same as the sum taken over I \ {i₀}, we conclude that

X

j6=i0

(λ_j −µ_j)D_j|D_i0

is pseudo-effective, too. Now all λ_j −cµ_j ≤0 and λ_j1 −cµ_j1 <0 with D_j1 ∩D_i0 6= ∅, hence the dual of

X

j6=i0

(λ_j −µ_j)D_j|D_i0

is effective non-zero, a contradiction.

Case 2 Now we deal with the case thatL6=D^′. Then we can write L =D^′+L₀

where L^m₀ ∈ Pic⁰(X) (The exponent m is there because there might be torsion in H²(X,Z); we take m to kill the denominator of the torsion part). We may in fact assume that m= 1; otherwise we pass to a finite ´etale cover ˜X of X and argue there (the vanishing on ˜X clearly implies the vanishing on X). Then the sequence S is modified to

(S) 0−→I(h)⊗(K_X +L)−→K_X +L−→(K_X+L)|D^′ = (K_D^′ +L₀)|D^′−→0.

Taking cohomology as before, things come down to prove

(∗) H⁰(D^′,−L₀|D^′) = 0.

If−L₀|D^′ 6= 0, then we see as above that −L₀|D^′ cannot have a nilpotent section. So if (∗) fails, then−L₀|D^′has a sectionssuch thats|redD^′has no zeroes, so that−L₀|redD^′ is trivial. But then −L₀|D^′ is trivial. Now let α:X −→A be the Albanese map with image Y. Then L₀ = α^∗(L^′₀) with some line bundle L^′₀ on A which is topologically trivial but not trivial. Since L₀|D^′ is trivial, we conclude that α(D^′)6= Y, and α(D^′) is contained in a proper subtorus B of A. Now consider the induced map

β :X −→A/B

and denote its image by Z. Then β(D^′) is a point; on the other hand D^′ is nef, so that dimZ = 1 and D^′ consists of multiples of fibers of β. But this contradicts D^′2 =L² 6= 0.

For applications to minimal K¨ahler 3-folds, 4.1 is still not good enough, because we need to know the vanishing propertyH²(X, mKX) = 0 on aQ-Gorenstein 3-fold (with K_X² 6= 0). We would like to set L = (m−1)K_X to apply 4.1 but this is no longer a line bundle. This difficulty is overcome by

4.3 Proposition. Let X be a normal Q-Gorenstein compact K¨ahler 3-fold with at most terminal singularities. Let A be a Q-line bundle. Suppose A is nef and A² 6= 0.

Then

H²(X, A+K_X) = 0.

Proof. (A) In a first step we show that we may assumeX to beQ-factorial. (Actually, in our application in Section 5, it will be clear that we may always assume X to be Q-factorial, so the reader only interested in the applications may skip (A)).

In fact, if X is not Q-factorial, there exists a bimeromorphic map f : Y −→ X from a normal Q-factorial K¨ahler space with at most terminal singularities ([Ka88,4.5^′]).

Moreover f is an isomorphism in codimension 1 and f is projective since X has only isolated singularities. Now consider the reflexive sheaf

H=f^∗(O_X(A))^∗∗. Choose a number r such that A^[r] is locally free. Then

H^[r] =f^∗(O_X(rA)),

since both sheaves are reflexive and coincide in codimension 1. ThusH is nef (asQ-line bundle) with H² 6= 0. Once we know the result in the Q-factorial case, we get

H²(Y,H⊗Kˆ _Y) = 0.

So by the Leray spectral sequence, we only have to show R¹f_∗(H⊗Kˆ _Y) = 0.

This however follows from [KMM87, 1-2-7]. Actually this citation deals with the al- gebraic case. However first notice that our statement is local around the isolated singularities of X. Now isolated singularities are algebraic by Artin’s theorem, i.e. we can realize an open neighborhood of an isolated singularity as an open set in a normal algebraic variety. So locally on X the map f : Y −→X can be realized algebraically.

Now we can approximate H by algebraic reflexive sheaves H_k up to high order k and then apply [KMM87, 1-2-7] to get the vanishing R¹f_∗(H_k) = 0. This sheaf coincides withR¹f∗(H) to high order, soR¹f∗(H) vanishes to high order. Fork approaching ∞, we obtain the vanishing we are looking for.

(B) From now on we assume X to be Q-factorial. We proceed as in the proof of 4.1.

First of all chooser such thatA^[r] is locally free. Then choose a singular metrich with positive curvature current onA^[r]. Now ¹_rh is a metric at least onA|X_reg with positive curvature current T extending to all of X. We argue as in the first part of the proof of 4.1 to obtain the divisor D and the current R, however D is only an integral Weil divisor. By the same arguments as in 4.1 we can still reduce the problem to proving

H²(D,O_D(A+K_X)) = 0.

(Notice that I_D ⊗O_X(A+KX) = O_X(−D+A+KX) outside a finite set and that by definition O_D(A+K_X) = O_X(A+K_X)|D)). Now D is Cohen-Macaulay; here we need in an essential way that locally X is the quotient of a hypersurface by a finite group. To be more detailed, we can write locally X = V /G with V a hypersurface singularity and G a finite group (see e.g. [Re87]). Let π : V −→ X be the quotient map and let ˆD = π^∗(D). If we can prove that ˆD is Cohen-Macaulay, then D will

§5. The caseK_X² = 0, KX 6= 0 17

be Cohen-Macaulay, too, since this property is G-invariant. So we may assume that X =V. Now X is (locally) a compound Du Val singularity [Re87], i.e. a 1-parameter deformation of a 2-dimensional rational double point. Hence we can find a Cartier divisorH ⊂X throughx₀ which has just a rational double point at x₀. Now consider D∩H. This is a Weil Q-Cartier divisor onH. Since x0 is a quotient singularity ofH, we can argue as above to see thatD∩H is Cohen-Macaulay. HenceDhas a hyperplane section through x0 which is Cohen-Macaulay. Thus D is Cohen-Macaulay at x0 itself.

Therefore we have by Serre duality

H²(D,O_D(A+K_X))≃ Hom(O_D(A+K_X),O_D(K_D)).

Suppose that H² does not vanish. Then we obtain a non-zero homomorphism s : O_D(A+K_X) −→K_D. This s must be generically non-zero. In fact, D is generically Gorenstein. Hence O_D(K_D)_x is isomorphic to an ideal in O_X,x for all x, in particular K_D has no torsion sections, D being Cohen-Macaulay; see [Ei95] in the algebraic case.

Let X0 be the regular part of X, this means that we eliminate a finite set fromX, all singularities being terminal. Let denote D₀ = D∩X₀ and let s₀ = s|X₀. Then by adjunction we have

06=s₀ ∈H⁰(D₀,O_D(−A+D)).

From now on we argue as in 4.1 just working on X0 instead ofX. The only exceptions are calculations of intersection numbers and Hodge index arguments. Here we still need to argue on X - we do not have any problems with singularities since all divisors are Q-Cartier.

§5. The case K

= 0, K

6= 0

The second ingredient for the proof of the abundance theorem for K¨ahler threefolds is the following weak analogue of 4.3 in caseK_X² = 0 (however one should have in mind that we are dealing with a case which does not exist a posteriori).

5.1 Theorem. Let X be a normal compact K¨ahler threefold with at most terminal singularities such that KX is nef. Suppose that K_X² = 0, KX 6= 0, and that X is simple and not Kummer. Then

h²(X, mKX)≤1 for all m∈N.

As already mentioned the essential property derived from X being simple non- Kummer is that π1(X) is finite [Ca94].

5.2 Start of the proof. Using Kawamata’sQ-factorialisation theorem (compare with the proof of 4.3), we may assume that X is Q-factorial. Suppose h²(X, mK_X) ≥ 2.

Using Serre duality we get – following Miyaoka and Shepherd-Barron – (many) non- split extensions

0−→K_X −→E−→mK_X −→0 (S)

with reflexive sheaves E of rank 2. We note

c₁(E) = (m+ 1)K_X (5.2.1)

and

c2(E) = 0. (5.2.2)

5.3 The unstable case.

(5.3.a) Here we will assume that every non-split extension E as in (S) is ω- unsta- ble for some fixed K¨ahler form ω independent of E. Let A_S ⊂ E be the ω-maximal destabilizing subsheaf. Then A_S is a Q-line bundle and we determine a Q-line bundle B_S such thatE/A_S =I_ZB_S with some subspace Z of codimension at least 2 (actually Z is generically (i.e. on the smooth part of X) locally a complete intersection or finite and supported in SingX). SinceK_X 6= 0, we obtain injective maps

φS :A_S −→mKX

and

ψ_S :K_X −→IB_S.

Now there are (up to C^∗) only finitely many maps φS : A_S −→ mKX with some Q- line bundle A_S arising as maximalω-destabilizing subsheaf for some extension (S). In fact, fix φ= φS : A −→mKX. Then by (6.13) there are only finitely many maximal reflexive subsheaves A^′⊂ mK_X such that A^′ 6⊂A. So we may suppose A^′ ⊂A. Then

A^′·ω² <A·ω². Actually, putting

ǫ:= minY_j ·ω²,

where the minimum runs over the finitely many irreducible hypersurfaces Yj ⊂X, we have

A^′·ω² ≤A·ω²−ǫ.

On the other hand, restricting ourselves to A^′ of the form A^′ = A_S′, we have by the destabilizing property

A^′·ω² ≥ c₁(E)·ω²

2 . (∗)

As there is an integral divisor P

λjYj such that A^′ =A⊗O_X(−X

λ_jY_j),

the finiteness of irreducible hypersurfaces inX gives the finiteness claim, since (∗) reads (A−X

λ_jY_j)·ω² ≥ c₁(E)·ω²

2 .

So we have only finitely many possible maps φ (up to C^∗). In the same way (by dualizing) we have only finitely many maps ψ (up to C^∗). In (5.3.b) below we prove that (φ, ψ) and (λφ, λψ) withλ ∈C^∗always define isomorphic extensions (S). Therefore

§5. The caseK_X² = 0, KX 6= 0 19

in total (λφ, µψ) withλ, µ∈C^∗ define just a 1-dimensional space of extensions, whence h²(X, mK_X)≤1.

(5.3.b)We shall now prove that the extension class defining (S) is already determined by φ andψ (modulo C^∗). So take another extension

0−→K_X −→E^′ −→mK_X −→0 (S^′)

with the same destabilizing sheaves A and B and with the same morphisms φ and ψ (the case of (φ, ψ),(λφ, λψ) is exactly the same). Let D be the divisorial part of

{φ= 0} ∪ {ψ = 0} ∪Sing(X);

then we obtain a splitting of the sequence (S) over X\D via φ: E≃mKX ⊕KX ≃A⊕B,

and an analogous splitting of E^′ over X \D. Observe also that A = mKX −D and B=K_X+D. Thus we obtain an isomorphism

f :E−→E^′

on X\D making the two extensions (S) and (S^′) isomorphic over X\D:

0 −→ K_X −→ E −→ mK_X −→ 0

k ↓ k

0 −→ K_X −→ E^′ −→ mK_X −→ 0.

It remains to extend the map f to X. Let us notice that we may assume Z = ∅.

In fact let Z1 be the codimension 1 part of Z. Restricting our two exact sequences describing E to D, we see that (modulo torsion at finitely many points)

A|D=K_X|D, hence

(m−1)K_X·D=D·D. (5.3.1)

In particular we note that D|D is nef, hence D itself is nef. Now (5.3.1) yields 0 =c₂(E) =c₁(A)·c₁(B) +c₂(I_ZB) = (mK_X −D)·(K_X +D) +Z₁ =Z₁, hence Z₁ =∅. In particular Z ⊂SingX has codimension at least 3.

This shows that we may ignore Z in all our following considerations; in what follows restriction will always that we also divide by torsion.

Take a local sections ∈E(U) over a small discU. We need to show that f(s)∈E^′(U);

a priori we only know f(s)∈E^′(D)(U). Let

κ:E→mK_X, κ^′ :E^′ →mK_X